The combinatorics of $\mathbb{C}^*$-fixed points in generalized Calogero-Moser spaces and Hilbert schemes

TL;DR

This paper explores the combinatorial structure of fixed points in cyclic quiver varieties related to rational Cherednik algebras, providing explicit classifications, character calculations, and new proofs of classical formulas.

Contribution

It classifies and constructs fixed points in cyclic quiver varieties, describes bijections between fixed points, and generalizes the q-hook formula using combinatorial methods.

Findings

Explicit classification of fixed points in cyclic quiver varieties

Calculation of characters of tautological bundles at fixed points

A new proof and generalization of the q-hook formula

Abstract

In this paper we study the combinatorial consequences of the relationship between rational Cherednik algebras of type $G(l,1,n)$, cyclic quiver varieties and Hilbert schemes. We classify and explicitly construct $\mathbb{C}^*$-fixed points in cyclic quiver varieties and calculate the corresponding characters of tautological bundles. Furthermore, we give a combinatorial description of the bijections between $\mathbb{C}^*$-fixed points induced by the Etingof-Ginzburg isomorphism and Nakajima reflection functors. We apply our results to obtain a new proof as well as a generalization of the $q$-hook formula.